Electronic structure of sodium tungsten bronzes ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ by high-resolution angle-resolved photoemission spectroscopy

The electronic structure of sodium tungsten bronzes, ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$, for full range of $x$ is investigated by high-resolution angle-resolved photoemission spectroscopy (HR-ARPES). The experimentally determined valence-band structure has been compared with the results of ab initio band-structure calculation. The HR-ARPES spectra taken in both the insulating and metallic phase of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ reveal the origin of metal-insulator transition (MIT) in the sodium tungsten bronze system. In the insulating ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$, the near-$E_F$ states are localized due to the strong disorder caused by the random distribution of ${\mathrm{Na}}^{+}$ ions in $\mathrm{W}{\mathrm{O}}_3$ lattice. While the presence of an impurity band (level) induced by Na doping is often invoked to explain the insulating state found at low concentrations, there is no signature of impurity band (level) found from our results. Due to disorder and Anderson localization effect, there is a long-range Coulomb interaction of conduction electrons; as a result, the system is insulating. In the metallic regime, the states near $E_F$ are populated and the Fermi level shifts upward rigidly with increasing electron doping $\left(x\right)$. The volume of electronlike Fermi surface (FS) at the $\Gamma \left(X\right)$ point gradually increases with increasing Na concentration due to $\mathrm{W}5d{t}_{2g}$ band filling. A rigid shift of $E_F$ is found to give a qualitatively good description of the FS evolution.

Figure 1

Schematic view of (a) optical properties and (b) phase diagram of the ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ system with Na concentration $\left(x\right)$. (c) Cubic crystal structure of $\mathrm{Na}\mathrm{W}{\mathrm{O}}_3$. (d) Rigid-band model for $\mathrm{W}{\mathrm{O}}_3$ showing the Fermi level $E_F$ in between the bottom of conduction band and the top of valence band. (e) Cubic Brillouin zone showing highly symmetric lines of $\mathrm{Na}\mathrm{W}{\mathrm{O}}_3$. (f) Geometrical arrangement for the ARPES measurements.

Figure 2

Valence-band HR-ARPES spectra of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.025$ measured along (a) $\Gamma \left(X\right)\text{−}X\left(M\right)$ and (b) $\Gamma \left(X\right)\text{−}M\left(R\right)$ directions at $130\mathrm{K}$. (c) Experimental valence-band structure obtained from HR-ARPES experiments at $130\mathrm{K}$. Bright areas correspond to the experimental bands. The theoretical band calculation of $\mathrm{W}{\mathrm{O}}_3$ after shifting $E_F$ is also shown by thin solid and dashed lines for comparison.

Figure 3

(a) Valence-band PES spectra of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.025$ at several temperatures showing the signature of polaron formation at the valence-band edge, $2.5\mathrm{eV}$. The inset shows the temperature dependence of the intensity ratio of the $2.5\mathrm{eV}$ peak to the $4.0\mathrm{eV}$ peak. (b) Intensity variation of $2.5\mathrm{eV}$ peak with respect to wave vector $k$ along the $\Gamma \left(X\right)\text{−}X\left(M\right)$ direction at 130 and $230\mathrm{K}$.

Figure 4

Typical valence-band HR-ARPES spectra of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.8$ measured at $14\mathrm{K}$ with $\mathrm{He}\mathrm{I}\alpha$ photons $\left(21.218\mathrm{eV}\right)$ along (a) $\Gamma \left(X\right)\text{−}X\left(M\right)$ and (b) $\Gamma \left(X\right)\text{−}M\left(R\right)$ directions. (c) Comparison of HR-ARPES spectra near valence-band edge in highly metallic ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.58$, 0.65, 0.7, and 0.8. The inset shows the calculated Fermi energy shift with Na doping $x$ from the rigid-band model. (d) Experimental valence-band structure of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.8$ obtained from HR-ARPES experiments along the highly symmetric directions. Bright areas correspond to the experimental bands. The theoretical band calculation of $\mathrm{Na}\mathrm{W}{\mathrm{O}}_3$ is also shown by thin solid and dashed lines for comparison.

Figure 5

HR-ARPES spectra near $E_F$ of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.025$ measured at $130\mathrm{K}$ along (a) $\Gamma \left(X\right)\text{−}X\left(M\right)$ and (b) $\Gamma \left(X\right)\text{−}M\left(R\right)$ directions. Vertical bars are a guide to eyes for band dispersion. (c) Experimental near-$E_F$ band structure along the highly symmetric directions. Theoretical band structure of $\mathrm{W}{\mathrm{O}}_3$ after shifting $E_F$ is also shown by thin solid and dashes lines for comparison.

Figure 6

Experimental near-$E_F$ HR-ARPES spectra along with the experimental band structure of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for (a) $x=0.58$, (b) $x=0.65$, (c) $x=0.7$, and (d) $x=0.8$ measured at $14\mathrm{K}$. Only the $\Gamma \left(X\right)$ region is shown from the $\Gamma \left(X\right)\text{−}M\left(R\right)$ direction of BZ. Vertical bars are a guide to eyes for band dispersion in EDCs. Theoretical band structure of $\mathrm{Na}\mathrm{W}{\mathrm{O}}_3$ is also shown in the intensity map by thin, solid, and dashed lines for comparison. Open circles show the highest intensity in experimental band mapping. (e) Near-$E_F$ momentum distribution curve (MDC) of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.8$. The gray solid line represents the MDC peak dispersion. The Fermi velocity $v_F$ is determined from the slope $(\delta E∕\delta k)$ at $E_F$. (f) The effective mass $(m^{*}∕m_e)$ with respect to Na concentration in ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$.

Figure 7

(a) Remnant Fermi surfaces of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.025$ at $130\mathrm{K}$ in the first BZ. Solid and dashed circles show the highest intensity points. (b) LEED pattern of (001) cleaved surface of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.025$ measured with primary energy of $54\mathrm{eV}$ at $130\mathrm{K}$. It shows the $1\times 1$ bulk and the $2\times 2$ surface superlattice spots. (c) LEED pattern of (001) cleaved surface of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for $x=0.8$ measured with primary energy of $106\mathrm{eV}$ at $14\mathrm{K}$. Fermi surfaces of ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ for (d) $x=0.58$, (e) $x=0.7$, and (f) $x=0.8$ showing electronlike pocket at the $\Gamma \left(X\right)$ point. Dotted lines around the $\Gamma \left(X\right)$ point are the calculated Fermi surface(s) (on $\Gamma XMX$ and $XMRM$ planes) for fractional Na concentration based on the rigid-band model.